## Always another problem

I was picking through the XKCD archives the other day and came across this comic, which stands by itself as being entirely true and hilarious. But all XKCD comics have a second part, a second punch-line, listed in the image’s “title” text.

This particular comic’s title text read as follows: “I first saw this problem on the Google Labs Aptitude Test. A professor and I filled a blackboard without getting anywhere. Have fun.”

This struck me as one of those nerd experiences that Randal Munroe, author of XKCD, occasionally writes about, that would be really cool to actually experience. To have a teacher so cool that not only would he would take the time to lend his expertise to help solve some purposely inscrutable problems, and spend quite a while at it. But after contemplating this for a few moments, I realized that it already had happened to me. Well, sort of.

See, many years ago, in the fall of 2004, a billboard appeared in Silicon Valley that said “{the first 10-digit prime found in consecutive digits of e}.com” and word of this reached my high school sophomore ears, during Linux class.

Tabling that week’s lesson — it was a some-what self-paced course — for a later day, I began to crack in to this problem with the kind of zeal shown by the physicist in the above comic. It made sense to solve the problem in Python, which I had been teaching myself, and the Fedora Core Linux computers in the class room had it already installed, so I just opened a vi window and started in.

That part only took a little while, because Python’s string slicing made iterating through an arbitrarily long string (stored in a text file from one of these websites that has e to the millionth digit) a piece of cake. Mix in a function that probabilistically test for primes, and you’re done.

But, of course, this was just the beginning. After chewing its cud for a while, my program spit out “7427466391”, which took me to 7427466391.com. That, of course, had another problem. This one was much trickier:

f(1) = 7182818284

f(2) = 8182845904

f(3) = 8747135266

f(4) = 7427466391

f(5) = ???

Just looking at the provided results, I could tell that it wasn’t any kind of conventional f(x) = kx function, or even any function there f(x) varied directly or indirectly with x. Aside from that, my math schooling through Honors Geometry provided very little help.

But at this point, some of the rest of the class began pitching in. See, somewhere along the way, while I had been solving the first part, I guess some of my classmates had noticed the, uh, *concentrated* look on my face, and wondered what I was up to. Stumped at this second puzzle, I opened the floor for input. A number of theories were floated and eventually sunken, to no avail.

Eventually, the teacher of this Linux class, one of the best teachers in the whole school, became interested in what was taking up so much attention. After explaining what was going on in full, and repeatedly reinforcing the fact that this was being done *in Linux* and was undoubtedly scholarly, he tacitly admitted the project’s merit and began providing assistance.

From there, I don’t remember how exactly the algorithm for finding the correct answer was determined, although I’m sure it was implemented in the same Python used on the first problem. But someone eventually figured out that the numbers were ten-digit sequences of numbers from e whose digits summed to 49. Once the fifth such number was found, and provided to this challenge website, it informed us that this was, in fact, a test.

As the message from Google Labs read:

One thing we learned while building Google is that it’s easier to find what you’re looking for if it comes looking for you. What we’re looking for are the best engineers in the world. And here you are.

As you can imagine, we get many, many resumes every day, so we developed this little process to increase the signal-to-noise ratio.

The sense of accomplishment seeing this was tempered only by the disappointment at having no resume to send given that I hadn’t yet held my first job, much less still being in the midst of achieving a diploma.

I must be overlooking something important about the “grid-of-resistors” question, because it seems rather simple to me. Electricity will follow the path of least resistance, right? In this case, there are three such paths, and they all have the same resistance: 3 ohms. Why isn’t that the answer?

As I understand it, electricity will take as many paths as possible. Not only would the three-step paths be taken, but the four-step and the five-step, ad infinitum.

Limits converging to infinity would be involved.